Complex nearly immotile behaviour of enzymatically driven cargos.

We report a minimal microtubule-based motile system displaying signatures of unconventional diffusion. The system consists of a single model cargo driven by an ensemble of N340K NCD motors along a single microtubule. Despite the absence of cytosolic or cytoskeleton complexity, the system shows complex behavior, characterized by sub-diffusive motion for short time lag scales and linear mean squared displacement dependence for longer time lags. The latter is also shown to have non-Gaussian character and cannot be ascribed to a canonical diffusion process. We use single particle tracking and analysis at varying temperatures and motor concentrations to identify the origin of these behaviors as enzymatic activity of mutant NCD. Our results show that signatures of non-Gaussian diffusivities can arise as a result of an active process and suggest that some immotility of cargos observed in cells may reflect the ensemble workings of mechanochemical enzymes and need not always reflect the properties of the cytoskeletal network or the cytosol.

A mathematical model and quantitative comparison of the small RNA circuit in the Vibrio harveyi and Vibrio cholerae quorum sensing systems.

Quorum sensing is the process by which bacteria regulate their gene expression based on the local cell-population density. The quorum sensing systems of Vibrio harveyi and Vibrio cholerae are comprised of a phosphorelay cascade coupled to a small RNA (sRNA) circuit. The sRNA circuit contains multiple quorum regulated small RNA (Qrr) that regulate expression of the homologous master transcriptional regulators LuxR (in V. harveyi) and HapR (in V. cholerae). Their quorum sensing systems are topologically similar and homologous thereby making it difficult to understand why repression of HapR is more robust than LuxR to changes in Qrr. In this work we formulate and parameterize a novel mathematical model of the V. harveyi and V. cholerae sRNA circuit. We parameterize the model by fitting it to a variety of empirical data from both species. We show that we can distinguish all of the parameters and that the parameterizations (one for each species) are robust to errors in the data. We then use our model to propose some experiments to identify and explain kinetic differences between the species. We find that V. cholerae Qrr are more abundant and more sensitive to changes in LuxO than V. harveyi Qrr and argue that this is why expression of HapR is more robust than LuxR to changes in Qrr.

The effect of divalent vs. monovalent ions on the swelling of mucin-like polyelectrolyte gels: governing equations and equilibrium analysis.

We introduce a comprehensive model of a mucin-like polyelectrolyte gel swelling-deswelling which includes the ion-mediated crosslinking of polymer strands and the exchange of divalent and monovalent ions in the gel. The gel is modeled as a multi-phase mixture which accounts for the polymer and solvent volume fractions and velocities as well as ionic species concentrations. Motion is determined by force balances involving viscous, drag, and chemical forces. The chemical forces are derived from a free energy which includes entropic contributions as well as the chemical and electrostatic interactions among the crosslinked polymer, uncrosslinked polymer, and the ionic solvent. The unified derivation produces all the classical effects (van't Hoff osmotic pressure, Donnan equilibrium potential, Nernst-Planck motion of ions) as well as expressions for Flory interaction parameter and the standard free energy parameters that explicitly depend on the gel chemistry and crosslink structure. For this model, we show how the interplay between ionic bath concentrations, ionic binding, and transient divalent crosslinking leads to a variety of swelled and deswelled phases/phase transitions. In particular, we show how the absorption of divalent ions can lead to a massive deswelling of the gel. We conclude that the unique properties of mucin-like gels can be explained by their ionic binding affinities and transient divalent crosslinking.

A molecular ruler mechanism for length control of extended protein structures in bacteria.

The lengths of the hook structure of flagellar motors and of the needle of the injectosome are both carefully controlled, by apparently similar mechanisms. In this paper we propose a novel mechanism for this length control and develop a mathematical model of this process which shows excellent agreement with published data on hook lengths. The proposed mechanism for length control (described using biochemical nomenclature appropriate for hooks) is as follows: Hook growth is terminated when the C-terminus of the length control molecule FliK interacts with FlhB, the secretion gatekeeper. The probability of this interaction is an increasing function of the length of the hook for two reasons. First, FliK is secreted through the hook intermittently during hook growth. Second, the probability of interaction with FlhB is a function of the amount of time the C-terminus of a secreted FliK spends in the vicinity of FlhB. This time is short when the hook is short because the folding of FliK exiting the distal end of the hook acts to pull the FliK molecule through the hook rapidly. In contrast, this time is much longer when the hook is longer than the unfolded FliK polymer since movement through the tube is not enhanced by folding. Thus, it is much more likely that interaction will occur when the hook is long than when the hook is short.

How Salmonella Typhimurium measures the length of flagellar filaments.

We present a mathematical model for the growth and length regulation of the filament of the flagellar motor of Salmonella Typhimurium. Under the assumption that the molecular constituents are translocated into the nascent filament by an ATPase and then move by molecular diffusion to the growing end, we find a monotonically decreasing relationship between the speed and the velocity of growth that is inversely proportional to length for a large length. This gives qualitative but not quantitative agreement with data of the velocity of growth. We also propose that the length of filaments is "measured" by the rate of secretion of the sigma(28)-antifactor FlgM, using negative feedback, and present a mathematical model of this regulatory network. The combination of this regulatory network with the length-dependent rate of growth enable the bacterium to detect length shortening and regrow severed flagellar filaments.

A model for length control of flagellar hooks of Salmonella typhimurium.

We present a mathematical model for the growth and length regulation of the hook component of the flagellar motor of Salmonella typhimurium. Under the assumption that the molecular constituents are translocated into the nascent filament by an ATP-ase and then move by molecular diffusion to the growing end, where they polymerize into the growing tube, we find that there is a detectable transition from secretion limited growth to diffusion limited growth. We propose that this transition can be detected by the secretant FliK, allowing FliK to interact with FlhB thereby changing the secretion target of the type III secretion machinery and terminating the growth of the hook.

Intrinsic H(+) ion mobility in the rabbit ventricular myocyte.

The intrinsic mobility of intracellular H(+) ions was investigated by confocally imaging the longitudinal movement of acid inside rabbit ventricular myocytes loaded with the acetoxymethyl ester (AM) form of carboxy-seminaphthorhodafluor-1 (carboxy-SNARF-1). Acid was diffused into one end of the cell through a patch pipette filled with an isotonic KCl solution of pH 3.0. Intracellular H(+) mobility was low, acid taking 20-30 s to move 40 microm down the cell. Inhibiting sarcolemmal Na(+)-H(+) exchange with 1 mM amiloride had no effect on this time delay. Net H(+)(i) movement was associated with a longitudinal intracellular pH (pH(i)) gradient of up to 0.4 pH units. H(+)(i) movement could be modelled using the equations for diffusion, assuming an apparent diffusion coefficient for H(+) ions (D(H)(app)) of 3.78 x 10(-7) cm(2) s(-1), a value more than 300-fold lower than the H(+) diffusion coefficient in a dilute, unbuffered solution. Measurement of the intracellular concentration of SNARF (approximately 400 microM) and its intracellular diffusion coefficient (0.9 x 10(-7) cm(2) s(-1)) indicated that the fluorophore itself exerted an insignificant effect (between 0.6 and 3.3 %) on the longitudinal movement of H(+) equivalents inside the cell. The longitudinal movement of intracellular H(+) is discussed in terms of a diffusive shuttling of H(+) equivalents on high capacity mobile buffers which comprise about half (approximately 11 mM) of the total intrinsic buffering capacity within the myocyte (the other half being fixed buffer sites on low mobility, intracellular proteins). Intrinsic H(+)(i) mobility is consistent with an average diffusion coefficient for the intracellular mobile buffers (D(mob)) of ~9 x 10(-7) cm(2) s(-1).

Facilitation of intracellular H(+) ion mobility by CO(2)/HCO(3)(-) in rabbit ventricular myocytes is regulated by carbonic anhydrase.

Intracellular H(+) mobility was estimated in the rabbit isolated ventricular myocyte by diffusing HCl into the cell from a patch pipette, while imaging pH(i) confocally using intracellular ratiometric SNARF fluorescence. The delay for acid diffusion between two downstream regions approximately 40 microm apart was reduced from approximately 25 s to approximately 6 s by replacing Hepes buffer in the extracellular superfusate with a 5 % CO(2)/HCO(3)(-) buffer system (at constant pH(o) of 7.40). Thus CO(2)/HCO(3)(-) (carbonic) buffer facilitates apparent H(+)(i) mobility. The delay with carbonic buffer was increased again by adding acetazolamide (ATZ), a membrane permeant carbonic anhydrase (CA) inhibitor. Thus facilitation of apparent H(+)(i) mobility by CO(2)/HCO(3)(-) relies on the activity of intracellular CA. By using a mathematical model of diffusion, the apparent intracellular H(+) equivalent diffusion coefficient (D(H)(app)) in CO(2)/HCO(3)(-)-buffered conditions was estimated to be 21.9 x 10(-7) cm(2) s(-1), 5.8 times faster than in the absence of carbonic buffer. Facilitation of H(+)(i) mobility is discussed in terms of an intracellular carbonic buffer shuttle, catalysed by intracellular CA. Turnover of this shuttle is postulated to be faster than that of the intrinsic buffer shuttle. By regulating the carbonic shuttle, CA regulates effective H(+)(i) mobility which, in turn, regulates the spatiotemporal uniformity of pH(i). This is postulated to be a major function of CA in heart.

Diffusion induced oscillatory insulin secretion.

Oscillatory secretion of insulin has been observed in many different experimental preparations. Here we examine a mathematical model for in vitro insulin secretion from pancreatic beta cells in a flow-through reactor. The analysis shows that oscillations result because of an important interplay between flow rate of the reactor and insulin diffusion. In particular, if the ratio of flow rate to volume of the reaction bed is too large, oscillations are eliminated, in contradiction to the conclusions of Maki and Keizer (L. W. Maki and Keizer J. Mathematical analysis of a proposed mechanism for oscillatory insulin secretion in perifused HIT-15 cells. Bull. Math. Biol., 57(1995), 569-591). Furthermore, with reasonable numbers for the experimental parameters and the diffusion of insulin, the model equations do not exhibit oscillations.

A mathematical model for quorum sensing in Pseudomonas aeruginosa.

The bacteria Pseudomonas aeruginosa use the size and density of their colonies to regulate the production of a large variety of substances, including toxins. This phenomenon, called quorum sensing, apparently enables colonies to grow to sufficient size undetected by the immune system of the host organism. In this paper, we present a mathematical model of quorum sensing in P. aeruginosa that is based on the known biochemistry of regulation of the autoinducer that is crucial to this signalling mechanism. Using this model we show that quorum sensing works because of a biochemical switch between two stable steady solutions, one with low levels of autoinducer and one with high levels of autoinducer.

Elimination of spiral waves in cardiac tissue by multiple electrical shocks.

We study numerically the elimination of a spiral wave in cardiac tissue by application of multiple shocks of external current. To account for the effect of shocks we apply a recently developed theory for the interaction of the external current with cardiac tissue. We compare two possible feedback algorithms for timing of the shocks: a "local" feedback algorithm 11 (using an external electrode placed directly on the tissue) and a "global" feedback algorithm 22 (using the electrocardiogram). Our main results are: application of the external current causes a parametric resonant drift similar to that reported in previous model computations; the ratio of the threshold of elimination of the spiral wave by multiple shocks to the threshold of conventional single shock defibrillation in our model for cardiac tissue is about 0.5, while earlier, less realistic models predicted the value about 0.2; we show that an important factor for successful defibrillation is the location of the feedback electrode and the best results are achieved if the feedback electrode or the ECG lead is located at the boundary (or edge) of the cardiac tissue; the "local" and the "global" feedback algorithms show similar efficiency.

The biphasic mystery: why a biphasic shock is more effective than a monophasic shock for defibrillation.

We demonstrate that a biphasic shock is more effective than a monophasic shock at eliminating reentrant electrical activity in an ionic model of cardiac ventricular electrical activity. This effectiveness results from early hyperpolarization that enhances the recovery of sodium inactivation, thereby enabling earlier activation of recovering cells. The effect can be seen easily in a model of a single cell and also in a cable model with a ring of excitable cells. Finally, we demonstrate the phenomenon in a two-dimensional model of cardiac tissue.

A numerical method for the solution of the bidomain equations in cardiac tissue.

A numerical scheme for efficient integration of the bidomain model of action potential propagation in cardiac tissue is presented. The scheme is a mixed implicit-explicit scheme with no stability time step restrictions and requires that only linear systems of equations be solved at each time step. The method is faster than a fully explicit scheme and there is no increase in algorithmic complexity to use this method instead of a fully explicit method. The speedup factor depends on the timestep size, which can be set solely on the basis of the demands for accuracy. (c) 1998 American Institute of Physics.

A biophysical model for defibrillation of cardiac tissue.

We propose a new model for electrical activity of cardiac tissue that incorporates the effects of cellular microstructure. As such, this model provides insight into the mechanism of direct stimulation and defibrillation of cardiac tissue after injection of large currents. To illustrate the usefulness of the model, numerical stimulations are used to show the difference between successful and unsuccessful defibrillation of large pieces of tissue.

Direct activation and defibrillation of cardiac tissue.

It is shown that inhomogeneity due to gap junctional resistance allows an excitable cable to be directly activated or defibrillated by the application of a large amplitude point stimulus of short duration. Multiple-scale analysis is used to derive a model equation describing the response of the excitable medium to stimuli, and this equation is analyzed to give quantitative estimates of the direct stimulus threshold and the defibrillation threshold. These estimates are shown to be in good agreement with experimental findings.

Rotating spiral waves created by geometry.

The Belousov-Zhabotinsky reagent and numerical simulations were used to show that under high-frequency stimuli, rotating spiral waves can be initiated in a homogeneous excitable medium in the vicinity of domain boundaries or inexcitable barriers with sharp corners.

Effects of high frequency stimulation on cardiac tissue with an inexcitable obstacle.

We study numerically the effects of high-frequency stimulation in excitable cardiac tissue with an inexcitable obstacle using a Fitz-Hugh Nagumo type model. We show that if the frequency of stimulation is sufficiently high and the size of the inexcitable obstacle is sufficiently large, then a re-entrant pattern can be initiated. We also show that with overdrive stimulation a re-entrant pattern can be removed provided there are no obstacles at which new re-entrant patterns are created.

Generation of reentry in anisotropic myocardium.

INTRODUCTION: We investigated numerically the effects of the rotation of fiber axis orientation through the myocardial wall on wave propagation. METHODS AND RESULTS: We show that because of this rotation and inherent discrete properties of myocardium, a premature stimulus can create unidirectional conduction block leading to reentry. CONCLUSION: The dynamics of the subsequent reentrant patterns are complicated by the presence of rotational anisotropy, as the center of reentry drifts, and the reentry terminates in finite time when it collides with the domain boundary.

An eikonal-curvature equation for action potential propagation in myocardium.

We derive an "eikonal-curvature" equation to describe the propagation of action potential wavefronts in myocardium. This equation is used to study the effects of fiber orientation on propagation in the myocardial wall. There are significant computational advantages to the use of an eikonal-curvature equation over a full ionic model of action potential spread. With this model, it is shown that the experimentally observed misalignment of spreading action potential "ellipses" from fiber orientation in level myocardial surfaces is adequately explained by the rotation of fiber orientation through the myocardial wall. Additionally, it is shown that apparently high propagation velocities on the epicardial and endocardial surfaces are the result of propagation into the midwall region and acceleration along midwall fibers before reemergence at an outer surface at a time preceding what could be accomplished with propagation along the surface alone.

The effects of discrete gap junction coupling on propagation in myocardium.

A modified cable theory for a bi-domain model of myocardium that incorporates the effect of gap junctions as discrete objects coupling cardiac cells is derived. The theory is shown to be in agreement with a number of experiments that cannot be explained using standard continuous cable theory, and resolves some apparent contradictions on failure of propagation in two-dimensional anisotropic tissue. In addition, some as yet untested predictions of the theory are mentioned.